The research of functional weak limit theorems for branching processes arises
a lot of interest since many years ago. It was firstly formulated for a BGW process by [1]} and proved by [2]} and [3]}. These results have been extended to another classes of branching processes. For instance, a wide literature exists around weak convergence results for branching processes with immigration (BPI) since the pioneer work by [4]}, see also [5]} and references therein. In this paper we focus our attention on a weak convergence theorem for a critical CBP with a random initial number of individuals  and assuming finite second order moment on the this initial value. A similar result was already established for a single CBP in [6]}, and for an array of CBPs in [7]}, by assuming fixed initial numbers of progenitors using infinitesimal generators results for their proofs. Inspired in the paper [5]} on BPI we will use limit theorems for random step processes
towards a diffusion process provided in [9]} to obtain an alternative proof. The scheme of it follows similar steps to the ones in [5]}. An important feature of a CBP is that the value of \(Z_n\)  conditioned on the knowledge of the previous generation, \(Z_{n-1}=k\) , is a random sum of random variables, namely \(\sum _{j=1}^{\phi _n(k)}X_{n-1,j}\) , instead of a non-random sum as in the case of a BPI. This leads to handle the proofs of each steps using conditioning arguments different from those used in [5]}.
